Subdivision Surface Strange face choice?
The solution to this problem was to add the correct supporting geometry.  After some fiddling, I have found something that produced a nice smooth output.  What one should look for when this type of problem appears, as suggested by the comments on the question, is NGons.  Although blender has "full" support for NGons, the subdivision surface does not always play nicely.  Although I did not completely get rid of the NGons, I reduced the vertex count and the result is quite nice (I think).  Here is the result: